Displacement of non-Newtonian compressible fluids in plane porous media flow

Displacement of non-Newtonian fluid in porous media is of paramount importance in the flow modeling of oil reservoirs. Although numerical solutions are available, there exists a need for closed-form solutions in simple geometries. Here we revisit and expand the work of Pascal and Pascal, who analyzed the dynamics of a moving stable interface in a semi-infinite porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of power-law behavior, assuming continuity of pressure and velocity at the interface, and constant initial pressure. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation considering the fluids compressibility, yields a set of nonlinear second-order PDEs. If the fluids have the same consistency index n, the equations can be transformed via a self-similar variable; incorporation of the conditions at the interface shows the existence of a compression front ahead of the moving interface. After some algebra, one obtains a set of nonlinear equations, whose solution yields the location of the moving interface and compression front, and the pressure distributions. The previous equations include integrals which can be expressed by analytical functions if "n" is of the form k/(k+1) or (2k-1)/(k+1), with "k" a positive integer. Explicit expressions are provided k=1, 2; for other values, results are easily obtained via recursive formulae. All results are presented in dimensionless form; the pressure distribution and interface positions are studied and discussed as a function of the self-similar variable for different values of the mobility and compressibility ratios.